A proximal-gradient inertial algorithm with Tikhonov regularization: strong convergence to the minimal norm solution

TL;DR

This paper introduces a proximal-gradient inertial algorithm with Tikhonov regularization that guarantees strong convergence to the minimal norm solution and achieves fast convergence rates for the objective function.

Contribution

It presents a novel algorithm with proven strong convergence to the minimal norm solution and optimal convergence rates under specific parameter settings.

Findings

Sequence converges strongly to the minimal norm solution.

Achieves an $ ext{O}(k^{-2})$ convergence rate for the objective function.

Demonstrates fast convergence of the objective value, velocity, and sub-gradient.

Abstract

We investigate the strong convergence properties of a proximal-gradient inertial algorithm with two Tikhonov regularization terms in connection to the minimization problem of the sum of a convex lower semi-continuous function $f$ and a smooth convex function $g$. For the appropriate setting of the parameters we provide strong convergence of the generated sequence $(x_k)$ to the minimum norm minimizer of our objective function $f+g$. Further, we obtain fast convergence to zero of the objective function values in a generated sequence but also for the discrete velocity and the sub-gradient of the objective function. We also show that for another settings of the parameters the optimal rate of order $\mathcal{O}(k^{-2})$ for the potential energy $(f+g)(x_k)-\min(f+g)$ can be obtained.